Large Schröder Number

Return the large Schröder number S(n) for paths of semilength n, with S(0)=1 and S(1)=2; n is capped at 1,000.

Description

Large Schröder numbers count a family of lattice paths allowing diagonal steps while remaining on one side of a diagonal.

Large Schröder numbers count a family of lattice paths allowing diagonal steps while remaining on one side of a diagonal. The large and small Schröder conventions differ.^[1]

The tool returns value. The indexing and output conventions below determine how to read these values.

When to use Large Schröder Number

Use this for the following task: Return the large Schröder number S(n) for paths of semilength n, with S(0)=1 and S(1)=2; n is capped at 1,000. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Large Schröder Number and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n according to the field table, then run the tool.
Read value. Preserve exact decimal strings and the stated indexing or coefficient order.
Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.

Input field	Meaning	Value type and bounds	Requirement
n	Path semilength (resource limit 1000)	integer; minimum 0; maximum 1000	Required

Handle n=0 separately, then advance a two-term BigInt recurrence and divide by n+1 exactly at each step.

S(0)=1; S(1)=2; (n+1)S(n)=3(2n−1)S(n−1)−(n−2)S(n−2)

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
value	Large Schröder number	string (integer/arbitrary)

json

{
  "input": {
    "n": 5
  },
  "output": {
    "value": "394"
  }
}

Limitations and assumptions

This returns the large Schröder count for semilength n, capped at 1000. It does not enumerate paths or return the small Schröder sequence.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Motzkin Number when you need its operation: Count Motzkin paths of length n exactly, with M(0)=M(1)=1 and a 1,000-index resource limit. Choose Fuss–Catalan Number for the complementary operation: Return the exact p-ary tree Fuss–Catalan number C(pn,n)/((p−1)n+1), with p≥2 and pn≤5,000 as a computation resource limit.

Background reference: OEIS A006318.

References

OEIS A006318 — OEIS Foundation

Similar or alternative tools

Motzkin Number
Count Motzkin paths of length n exactly, with M(0)=M(1)=1 and a 1,000-index resource limit.
Fuss–Catalan Number
Return the exact p-ary tree Fuss–Catalan number C(pn,n)/((p−1)n+1), with p≥2 and pn≤5,000 as a computation resource limit.